Tomography, as used here, is a general term describing various techniques for imaging one or more cross-sectional xe2x80x9cfocal plane(s)xe2x80x9d through an object. Since contiguous focal planes can describe a volumetric region, this definition also encompasses reconstruction of arbitrary volumes. Tomography typically involves forming projections of a region of interest using some type of penetrating radiation, e.g. x-rays, sound waves, particle beams, or products of radioactive decay, combined with application of a reconstruction technique. Tomography has been applied in diverse fields to objects ranging in size from microscopic to astronomical. X-ray tomography, for example, is commonly used to inspect solder joints for defects formed during fabrication of printed circuit assemblies.
Computed tomography (CT) is a computational technique for reconstruction of cross-sectional images formed by processes which can be described (or approximated) by line integrals. To see that x-ray absorption imaging falls into this category, note that the relative attenuation of a monochromatic beam of x-radiation as it passes through an absorbing material is given in the non-diffracting case by the following equation:       I          I      0        =      ⅇ          -              ∮                              σ            ⁡                          (              r              )                                ⁢                      ⅆ            r                              
Where:   I      I    0  
is the relative attenuation of the monochromatic beam of x-radiation, "sgr" is the linear absorption coefficient, and the integral is to be evaluated along the beam path r. Taking the natural log of both sides results in the following form:       -          ln      ⁡              (                  I                      I            0                          )              =      ∮                  σ        ⁡                  (          r          )                    ⁢              ⅆ        r            
A discrete approximation to this line integral is typically used in practice, as follows:       y    i    =                    ln        ⁡                  (                      I                          I              0                                )                    i        =                  ∑                  xe2x80x83                    ⁢              xe2x80x83            ⁢                        a          ij                ⁢                  x          j                    
Where: yi is a measured projection, aij is the path length of the intersection between ray i and voxel (a three-dimensional pixel) j, and xj reflects the effective linear absorption coefficient within that voxel.
Methods for CT reconstruction have been widely discussed in the literature and will be summarized only briefly here. Transform methods rely on discrete approximations to inversion of the Radon transform in one form or another. Often, the Fourier slice theorem is exploited. The Fourier slice theorem states that the Fourier transform of a parallel projection of an object gives a slice through the Fourier transform of the object. In direct inversion, the Fourier transforms of the projections are used to build up an estimate of the Fourier transform of the object, which is then interpolated and inverted. In filtered backprojection, the Fourier transforms of the projections are multiplied by an appropriate filter function, and then the resulting values projected back along the paths of the incoming rays (hence, the term backprojection). Although derived from identical starting points, the two approaches have different computational requirements and different behavior in practical implementations. Filtered backprojection is typically favored in most implementations today, although there is continued research interest in direct inversion.
In contrast to transform methods, which derive from the inverse Radon transform, series-expansion methods for tomographic reconstruction, also known as algebraic reconstruction methods, directly seek a solution to the discrete ray equations y=Ax , where A is a matrix and x is a single row matrix. Noise-limited projection data frequently result in an underdetermined and inconsistent set of equations, so approximate solutions must be sought. Additionally, the system of equations is typically too large for inversion or singular value decomposition to be attractive. Instead, iterative methods related to Kaczmarz""s xe2x80x9cmethod of projectionsxe2x80x9d are used, which are described in A. C. Kak and Malcolm Slaney, xe2x80x9cPrinciples of Computerized Tomographic Imaging,xe2x80x9d IEEE Press, 1988, 276-285 and incorporated by reference herein. Among the better known are Algebraic Reconstruction Techniques (ART) including several variants, such as additive and multiplicative ART, Simultaneous Iterative Reconstructive Technique (SIRT), and Simultaneous Algebraic Reconstruction Technique (SART), which are described in Y. Censor, 1983, xe2x80x9cFinite Series-Expansion Reconstruction Methods.xe2x80x9d Proc. IEEE 71:409-419 and incorporated by reference herein. The various methods differ in the criteria used to choose a solution, the updating rule they use, and in the manner in which they sequence over the available projection data.
Many implementations of CT treat a 3-dimensional object as made up of adjacent, 2-dimensional slices (as in computerized axial tomography or CAT scanning) and compute multiple 2-D tomographic reconstructions. Direct 3D reconstructions are also done, generally on the basis of Grangeat""s method or variations thereof.
Computed tomography has the potential for more accurate reconstructions than laminography or tomosynthesis. Indeed, using complete data (projections from all angles), CT can be shown to provide an optimal, band-limited approximation to an object of finite size. If the original object is itself discrete, CT reconstruction can often be shown to be an exact model of the object. Shadowing artifacts and loss of high-frequency components are dramatically reduced, compared to laminography or tomosynthesis. Highly accurate reconstructions come with a price, however. CT typically requires many projections and is computationally intensive. As a result, its use to date has largely been limited to areas (such as medicine) where multi-million dollar machines can be economically justified.
As described above, current CT technology is not optimal for many tasks, including the inspection of printed circuit boards. Modem circuit boards are generally double-sided and utilize mainly surface-mount technology (SMT) parts in a variety of package styles. Additionally, through-hole components or pin-grid-array (PGA) parts may be used in some instances. Visual inspection or automated optical inspection (AOI) are inadequate for these boards, since many SMT joints are hidden under the component body. Since components are mounted on both sides of the board, simple transmission radiography is also not adequate, and some form of cross-sectional imaging is required.
Additionally, for some solder joint types, multiple slices at different depths are required for adequate characterization. CT would be ideal in terms of image quality, but is seldom used due to the large number of projections required and the associated costs. True 3-D tomography using cone beam illumination, rather than axial tomography, is desirable for efficient use of x-ray photons, and because projections in the plane of the board are typically impractical due to the excessive absorption. Laminography and tomosynthesis are both currently used for PCB inspection, but suffer from poorer image quality than would be available with CT. As noted above, both techniques suffer from shadowing artifacts, since out-of-focus information is blurred but not actually removed from the reconstructed slices.
Present techniques for CT require all of the projection data to be available before reconstruction can begin. In some architectures, this can lead to large storage requirements and delays in processing (i.e., throughput problems). When each projection is a two-dimensional image, for example a 1024xc3x971024 element image digitized with 12-bit resolution, more than one megabyte of storage is required for each projection. Additionally, in some tomographic systems, collection of projections for the first region of interest may be interleaved with collection of projections for other regions of interest. As a result, the system may be required to store large amounts of projection data before reconstruction can begin.
A system method for computed tomographic imaging is described. The system and method allows for beginning the processing for the reconstruction with as few as one projection, thus eliminating the requirement that a complete set of projections is available before reconstruction is attempted.
In one embodiment, the method includes acquiring at least one projection but less than all projections to be used in reconstruction of an unknown object and processing the at least one projection for reconstruction of the unknown object. Processing the projection(s) may include revising an estimate of the unknown object.
Other systems, methods, features and advantages of the invention will be or will become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features and advantages be included within this description, be within the scope of the invention, and be protected by the accompanying claims.